Globular Cluster Formation at High Density: A model for Elemental Enrichment with Fast Recycling of Massive-Star Debris

0.31 MB·

by  Bruce G. Elmegreen

Globular Cluster Formation at High Density: A model for
Elemental Enrichment with Fast Recycling of Massive-Star Debris
Bruce G. Elmegreen1
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ABSTRACT
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The self-enrichment of massive star clusters by p-processed elements is shown
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J to increase significantly with increasing gas density as a result of enhanced star

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formation rates and stellar scatterings compared to the lifetime of a massive star.

] Considering the type of cloud core where a globular cluster might have formed,
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we follow the evolution and enrichment of the gas and the time dependence
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of stellar mass. A key assumption is that interactions between massive stars
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are important at high density, including interactions between massive stars and
p
- massive star binaries that can shred stellar envelopes. Massive-star interactions
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r should also scatter low-mass stars out of the cluster. Reasonable agreement
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as with the observations is obtained for a cloud core mass of ∼ 4×106 M⊙ and a
[ density of ∼ 2×106 cm−3. The results depend primarily on a few dimensionless

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parameters, including, most importantly, the ratio of the gas consumption time
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4 to the lifetime of a massive star, which has to be low, ∼ 10%, and the efficiency
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of scattering low-mass stars per unit dynamical time, which has to be relatively
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1 large, such as a few percent. Also for these conditions, the velocity dispersions
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. of embedded globular clusters should be comparable to the high gas dispersions
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0 of galaxies at that time, so that stellar ejection by multi-star interactions could
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cause low-mass stars to leave a dwarf galaxy host altogether. This could solve
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: the problem of missing first-generation stars in the halos of Fornax and WLM.
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Subject headings: Galaxies: star clusters — Galaxies: star formation — globular
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clusters: general
1. Introduction
Most globular clusters (GCs) in the Milky Way have two populations of stars in ap-
proximately equal proportion with a first generation (G1) relatively abundant in Oxygen
1IBM Research Division, T.J. Watson Research Center, 1101 Kitchawan Road, Yorktown Heights, NY
10598;[email protected]
– 2 –
compared to Sodium and a second generation (G2) the reverse. This bimodality is presum-
ably the result of contamination of the G2 stars by the products of p-process nucleosynthesis
in the G1 stars, particularly the reactions 22Ne to 23Na along with the simultaneous destruc-
tionof 16O at a temperature of 2×107 K (Denissenkov & Denissenkova 1990; Decressin et al.
2007a). Other elemental anti-correlations include Mg with Al (e.g., Carretta et al. 2014),
explained by Langer et al. (1993) and others as a result of high temperature (T > 7 ×107
K) proton-capture plus beta-decay that transforms 24Mg into 25Mg, 26Mg and 27Al. Also in
some GCs, stellar Nitrogen anti-correlates with Carbon and Oxygen (Dickens et al. 1991),
with a total C+N+O abundance that is about constant, suggesting a CNO cycle. Reviews of
these abundance anomalies are in Gratton et al. (2004), Charbonnel (2005), Gratton et al.
(2012), Renzini et al. (2015), and Bastian (2015).
Light element anomalies from the CNO cycle and the NeNa- and MgAl-chains are pe-
culiar to the GCs and are not in field or halo stars (Gratton et al. 2000; Charbonnel 2005;
Prantzos & Charbonnel 2006). Because they involve high-temperature reactions and the
stars in which they are observed today are too low in mass to have had such p-process re-
actions themselves (Gratton et al. 2001), the Na-excess and Al-excess stars in current GCs
had to form in gas that was pre-enriched with these elements (Cottrell & Da Costa 1981),
most likely from the previous generation of stars as mentioned above. The debris from this
previous generationmay also have mixed with some left-over initial gasto explain ananticor-
relation between Li, which is destroyed in stars, and Na, which is produced (Pasquini et al.
2005; Bonifacio et al. 2007; Decressin et al. 2007a), and to explain the difference between the
high enrichment in the nuclear burning regions of massive stars and the observed enrichment
in low mass stars (Decressin et al. 2007b, table 4).
Thefirstgenerationhasbeenproposedtoincludenormalmassivestars(Cottrell & Da Costa
1981), such as rapidly spinning massive stars (Prantzos & Charbonnel 2006), which, be-
cause of their rotation, bring p-processed material from the H-burning zone into the en-
velope and then shed it along the equator via centrifugal force and radiation pressure
(Prantzos & Charbonnel 2006; Decressin et al. 2007b). Binary massive stars with Roche
lobe overflow are also an option (de Mink et al. 2009). AGB stars (D’Ercole et al. 2008)
shed processed gas at low speed too and may be a source of the anomalies, although the
near-constant C+N+O doesnot looklike abyproduct ofAGBstars, which make carbondur-
ing Helium burning (Charbonnel 2005); NGC 1851 may be an exception though (Yong et al.
2014; Simpson et al. 2017). Renzini et al. (2015) discuss ways in which the AGB option
might still be viable. Other models for the bimodality include proto-stellar disk accretion
(Bastian et al.2013a), supermassive stars(Denissenkov & Hartwick 2014;Denissenkov et al.
2015), GC-merging in dwarf galaxy hosts (Bekki & Tsujimoto 2016), and AGB wind re-
collection in the pressurized cavity around the GC (D’Ercole et al. 2016).
– 3 –
A problem with these models is that the stellar debris is only a small fraction of the
total stellar mass in a normal IMF. This implies that because the current G1 and G2 masses
are comparable to each other, the mass in the original G1 population had to be at least
the inverse of this fraction times the current G1 mass. For the model with rapidly rotating
massive stars and a normal IMF, the original cluster had to be ∼ 20 times the current G1
mass (Decressin et al. 2007b), and for the AGB model, the original cluster had to be > 10
times the current mass (D’Ercole et al. 2008). The missing 90%-95% of the G1 stars had
to escape, and even more had to escape if G2 stars escaped too, which is likely. These
escaping stars presumably comprise the halo of the Milky Way (Prantzos & Charbonnel
2006; Martell, et al. 2011), but such large halo amounts are not evident in the Fornax or
WLM dwarf galaxies (Larsen et al. 2012, 2014; Elmegreen et al. 2012).
Another constraint on the models is that there is virtually no elemental contamination
from G1 supernova inside the G2 stars (Charbonnel 2005; Renzini 2013). In the model
with rapidly rotating massive stars, this constraint implies that the G2 stars formed quickly,
before the G1 supernovae exploded. A potential problem here is that without supernovae,
the gas that formed G1 may not be cleared away from a massive cluster (Ginsburg et al.
2016; Lin et al. 2016), causing a high fraction of it to turn into stars. Such high efficiency
prevents the cluster from shedding its excess G1 stars during rapid gas loss (Krause et al.
2016), and then the final cluster is too massive and it has too many G1 stars compared
to G2. Cluster clearing was tested by Bastian et al. (2014b), who found that clusters with
masses up to 106 M and ages between 20 and 30 Myr are often clear of gas. This age is
⊙
old enough for some supernovae to have occurred. Without gas, star formation stops and a
prolonged epoch of secondary star formation that feeds on G1 debris does not occur.
Secondary or delayed star formation as in the AGB model is also not observed in to-
day’s massive clusters at the predicted age range of 10-1000 Myr (Bastian et al. 2013b;
Cabrera-Ziri et al. 2014), although it was reported by Li et al. (2016) whose result was then
questioned by Cabrera-Ziri et al. (2016). Neither do massive 100 Myr old clusters have
obvious gas from accumulated winds or secondary accretion (Bastian & Strader 2014).
Here we investigate further the model by Prantzos & Charbonnel (2006) and others,
where p-process contamination from massive stars is quickly injected into a star-forming
cloud core and incorporated into other forming stars. We consider conditions appropriate
for the early Universe where the interstellar pressure was much higher than it is today. This
pressure follows from the observation that star formation was ∼ 10 times more active per
unit area than it is now (Genzel et al. 2010) as a result of a factor of ∼ 10 higher gas column
density (Tacconi et al. 2010; Daddi et al. 2010). Because interstellar pressure scales with
the square of the column density (P ∼ GΣ2), the pressure was ∼ 100× larger when GCs
– 4 –
formed than it is in typical star-forming regions today. Consequently, the core of the GC-
forming cloud was likely to have a higher density and therefore a larger number of free-fall
times, such as 10 or more, before the first supernovae occurred. Star formation could have
occurred so quickly and the energy dissipation rate at high density could have been so high
that feedback from massive stars had little effect on cloud dispersal before the supernova
era (e.g., Wunsch et al. 2015). Also at high density, massive stars experience a significant
drag force from gas and low-mass stars (Ostriker 1999), causing them to spiral into an
even more compact configuration. High stellar densities lead to the dispersal of extrusion
disks filled with stellar envelope material (Prantzos & Charbonnel 2006), and also to close
encounters between massive stars or between tight massive binaries and massive stars, which
can shred the stellar envelopes (Gaburov et al. 2010). High stellar densities may also make
supermassive stars by coalescence (Ebisuzaki et al. 2001; Bally & Zinnecker 2005), and these
stars can produce the highest-temperature p-process elements (Denissenkov et al. 2015).
A related point is that at high density, a significant fraction of early-forming low-mass
stars should have been ejected from the cloud core by interactions with massive binaries
and massive multi-body collisional systems. This could occur long before “infant mortality,”
when thefinalclearing of gassignalstheend ofthestarformationprocess. Rapidandcontin-
uous stellar ejections by massive star interactions are well observed in numerical simulations
ofcluster formation(Reipurth & Clarke 2001;Bate & Bonnell 2005;Fujii & Portegies Zwart
2013). Because massive three-body collisions can also dump nearly a supernova’s worth of
kinetic energy into a region (Gaburov et al. 2010; Umbreit et al. 2008), the gas in the core
should be continuously agitated. The corresponding changes in the gravitational potential
energyofthegasshouldejectevenmorelow-massstars, inanalogytotheproposedejectionof
stars and dark matter from the cores of young star-bursting dwarf galaxies (Governato et al.
2012; El-Badry et al. 2016).
Dense cores are likely to continue accreting from the cloud envelope for many core
dynamical times. If the core plus envelope gas mixes with stellar debris and forms new stars,
then a succession of stellar populations will occur with ever-increasing levels of p-process
elements. By the time the first massive stars begin to supernova and clear away the gas,
some 3-7 Myr after star formation begins (Heger et al. 2003), there should be a wide range
of p-process elements in the stars that have formed. The model presented below shows that
this range can reproduce the observations for certain values of dimensionless parameters.
AkeyobservationisthatelementalenrichmentinGCsseemstobediscrete(Marino et al.
2011; Carretta et al. 2012, 2014; Renzini et al. 2015). Discreteness requires burst-like con-
tamination, and in the stellar interaction model here, that means intermittent events of
catastrophic massive-star interactions. The interactions occur where the density is high-
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est, so either the stellar density in the cloud core varies episodically with a burst of stellar
collisions following each high-density phase, or there are several mass-segregated cores in a
proto-GC(McMillan et al. 2007; Fujii & Portegies Zwart 2013) andeach has itsown burst of
intense interactions. Both situations are likely and could contribute to discrete populations
of stars. The temporal density variations would presumably follow from the time-changing
gravitational potential in the cluster core, as stellar envelope mass is disbursed along with
cloudmassthroughcollisions, andthenrecollected inthecoreafteradynamical timebecause
of self-gravity and background pressure.
The following sections examine this model in more detail. Section 2.1 outlines the basic
model of GC formation at high density, section 2.2 presents the equations that govern this
model, giving some analytical solutions in section 2.3, and section 3 shows the results. A
conclusion that highlights the main assumptions of the model and their implications for
high-density cluster formation is in Section 4.
2. Star Formation in Stellar Debris
2.1. Basic Model
The basic scale of GC formation considered here involves a molecular cloud core of mass
∼ 4 × 106 M in a spherical region of radius ∼ 3 pc. A larger region of lower-density gas
⊙
should surround this core, possibly in the form of filaments or spokes which continuously
deliver new gas (Klessen et al. 1998; Myers 2009). Perhaps the total mass involved with
the proto-GC and its neighborhood is ∼ 107 M or more, as observed for a massive dense
⊙
region in the Antenna galaxy (Herrera et al. 2012; Johnson et al. 2015). The core molecular
density for the above numbers is 2.2×106 cm−3 and the free fall time (= (3π/32Gρ)0.5) is
0.03 Myr. The ratio of the core mass to the free fall time, 133 M yr−1, is a measure of
⊙
the core accretion rate during core formation. If this core collapsed from the interstellar gas
at the typical rate of σ3 /G for interstellar velocity dispersion σ , then σ = 82 km
ISM ISM ISM
s−1, which is not unusual for high-redshift disk galaxies (e.g., F¨orster Schreiber, et al. 2009).
The accretion rate of low-density peripheral gas on to the core should be much less than the
initial core formation rate.
The fiducial cloud core mass of ∼ 4 × 106 M was chosen to produce a final GC
⊙
mass of around 2 × 105 M , which is at the peak of the GC mass distribution function
⊙
(Harris & Racine 1979). The first factor of ∼ 10 in mass reduction follows from our model
including multiple generations of star formation in the core and mass loss from stellar ejec-
tion, as required by the observed spread in p-processed elemental abundance. Another
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factor of ∼ 2 reduction in the GC mass is likely from evaporation over a Hubble time
(McLaughlin & Fall 2008).
Theaverage coresurfacedensity inthisbasicmodel, 1.4×105 M pc−2, iscomparableto
⊙
the maximum for stellar systems (Hopkins et al. 2010; Walker et al. 2016). The core velocity
dispersion is ∼ 76 km s−1, which is a factor of 2 higher than for massive clusters today, but
not unreasonable for a young galaxy where the gas turbulence speed is this high. Over time,
the cluster should expand and the dispersion decrease (e.g. Gieles & Renaud 2016). The
original cluster dispersion is high enough to make feedback-driven gas loss difficult before
the supernova era (Matzner & Jumper 2015; Krause et al. 2016). We consider that because
of this difficulty, the efficiency of star formation per unit free-fall time might be relatively
high, ∼ 10%, instead of the usual 1% (Krumholz & Tan 2007). Then the gas consumption
time, which is the free-fall time divided by the efficiency, is 0.3 Myr. The significant point
here is that this consumption time is much less than the evolution time of a high-mass star.
One potential implication of the high velocity dispersion in GC-forming cloud cores is
that stars ejected by time-changing gravitational potentials (e.g., binary or multi-star inter-
actionsandtheinduced rapidgasmotions) should alsohave fairlyhighvelocities. Dynamical
ejectionprocessescanbemuchmoreenergeticthanthermalevaporationfromarandomwalk.
This offers an intriguing solution to the problem stated in the introduction that Fornax and
WLM do not have stellar halos massive enough to include all of the required mass of G1
stars that should have been ejected from their GCs. In fact these are dwarf galaxies with
very slow internal motions: the Fornax dwarf spheroidal galaxy has a ∼ 12 km s−1 internal
velocity dispersion and a much lower rotation speed (Walker et al. 2006), while WLM has
a 36 km s−1 rotation speed (Leaman et al. 2012). Stars that are ejected at a factor of 1.5
to 2 times the escape speed of the GC could leave the galaxy. This possibility leads to the
prediction that galaxies with slow internal motions should have a systematic depletion in
halo stars from the G1 population that escaped their GCs. A similar conclusion was reached
by Khalaj & Baumgardt (2016) based on stellar loss from GCs during gas expulsion.
The IMF for all star formation is assumed to be fully populated and given by the log-
normal distribution in Paresce & De Marchi (2000) for stellar mass 0.1 M < M < 0.8 M ,
⊙ ⊙
with mass at the peak M = 0.33 M and dispersion σ = 0.34 M , and by a power
C ⊙ ⊙
law with the Salpeter slope −2.35 above 0.8 M (see also Prantzos & Charbonnel 2006).
⊙
The upper limit to the stellar mass will be varied from M = 100 M to 300 M ,
upper ⊙ ⊙
with the high value considered because of stellar coalescence. Note that a 320 M star
⊙
has been suggested for the dense massive cluster R136 in the LMC (Crowther et al. 2010;
Crowther et a. 2016), and starsmore massive than100 M were found inthe dense cluster in
⊙
NGC 5253 (Smith et al. 2016). Stars with masses larger than 20 M are assumed to make p-
⊙
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processelements (Decressin et al.2007a)andmixthemintotheir stellar envelopes, which are
definedtobeallofthestellarmassoutsideoftheHecore, asgivenbyPrantzos & Charbonnel
(2006). For M = 100 M , the fraction of the total stellar mass in the form of these
upper ⊙
envelopes is f = 7.9%, which is the product of the fraction of the IMF in stars with
env
M > 20 M (12.1%) and the average fraction of this stellar mass in the form of envelopes
⊙
(65.1%). For M = 300 M , f = 9.3% (16.4% of the IMF is in M > 20 M stars and
upper ⊙ env ⊙
56.8% of that mass is in envelopes). Also for these two upper masses, the fraction of the
total stellar mass in long-lived, low-mass stars (M < 0.8 M ), is f = 31.2% and 29.7%,
⊙ LM
respectively. The formation of p-process elements and the delivery of these elements into the
stellar envelopes and equatorial disks is assumed to proceed at a steady rate with an average
timescale t = 3 Myr, the lifetime of a high-mass star.
evol
Regarding the contamination by p-processed elements, we note that the [O/Na] ratio
in GCs varies from the large value of ∼ 0.4 in G1 to the small value of ∼ −1.4 in G2,
depending on the GC (−1.4 is for NGC 2802; Carretta 2006; Prantzos & Charbonnel 2006;
Gratton et al. 2012). The maximum value is about the same as in field stars and the min-
imum value is close what is expected in the envelope of a massive star near the end of its
pre-main sequence phase (Prantzos & Charbonnel 2006). The range of observed values, 1.8
dex, is less than the change expected inside the nuclear burning regions of massive stars,
which is 2.8 to 3.4 dex (depending on reaction rates) in Decressin et al. (2007a). This differ-
ence allows for some dilution of the core region with the envelope of the star before dispersal.
The full range in today’s stars therefore extends from the presumed initial condition when
the first generation formed, to a value that represents the near-complete conversion of a
massive stellar envelope into one or more G2 stars. Values of [O/Na] between these ex-
tremes correspond to some combination of incomplete mixing of the processed debris with
first generation gas, and partially processed gas from stars that have not yet finished their
main sequence evolution (Scenarios II and I in Prantzos & Charbonnel 2006).
The dilution proportions of processed stellar envelope gas and original cloud gas has
also been estimated from the relative Li abundance in the G2 stars, considering that Li will
be destroyed in the massive G1 stars. The observations suggest that up to 70% of the mass
of a G2 star could come from the debris of G1 stars (Decressin et al. 2007b). This high
fraction limits the amount of accretion from the cloud envelope during the star formation
process.
As mentioned above, the enriched gas is assumed to come from stellar equatorial disks
(Prantzos & Charbonnel 2006) and stellar debris generated by close interactions. For ex-
ample, Gaburov et al. (2010) show that massive binaries that collide with another star can
merge and puff up (see also Fregeau et al. 2004), and that continued collisions with the
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third star before merging can shred the common envelope. Mass loss fractions of ∼ 10%
are feasible in this situation. Umbreit et al. (2008) also consider the implications of col-
lisions and suggest that the kinetic energy from collision-induced mass loss can clear gas
away every few million years from aging globular clusters. This kinetic energy is comparable
to that of a supernova (Gaburov et al. 2010). For the dense cloud cores considered here,
that is not enough energy to clear away the gas, which is highly dissipative, but it could be
enough for later stages of cluster formation after a significant amount of gas has turned into
stars, and that is also when the supernovae themselves begin to clear the clusters of residual
gas. We assume here that all of the stellar debris, i.e., from massive-star equatorial disks
(Prantzos & Charbonnel 2006), massive star collisions, andRochelobe overflow aroundmas-
sive binaries(de Mink et al.2009), bringsp-processedmaterial fromtheenvelopes ofmassive
stars into the dense cloud core where it mixes with existing and newly accreted core gas on
the turbulent crossing time (∼ 0.1 Myr), and is incorporated into new stars. This process
continues for ∼ 3 Myr, forming the whole final cluster, at which point supernovae begin to
remove the remaining gas.
2.2. Model Equations
Consistent with these assumptions, we consider an initial cloud core of mass M (t = 0)
gas
in which stars begin to form, and a continuous accretion of new cloud gas onto this core at
a rate R . Stars are assumed to form in the core with a constant consumption time,
acc
t (equal to t divided by the efficiency per free fall time). The effect of varying
consume ff
the consumption time will be discussed below. The formation rates of low (< 0.8 M ),
⊙
intermediate (0.8 M − 20 M ) and high (> 20 M ) mass stars are given by the star
⊙ ⊙ ⊙
formationrateinthecoremultiplied bythefractionsoftheIMFinthesethreemassintervals:
dM /dt = f M (t)/t (1)
star,LM LM gas consume
dM /dt = f M (t)/t (2)
star,IM IM gas consume
dM /dt = f M (t)/t , (3)
star,HM HM gas consume
where f = 0.312, f = 0.567, and f = 0.121 for an IMF with a most massive star of
LM IM HM
100 M , and where f = 0.297, f = 0.540, and f = 0.164 for an IMF with a most
⊙ LM IM HM
massive star of 300 M (Sect. 2.1). The total mass formed in these stars is
⊙
t
M (t) = M˙ dt,... (4)
star,LM star,LM
Z0
and so on for the other mass ranges. Here we use the notation M˙ = dM/dt. The total mass
for all stars is M = M +M +M .
star star,LM star,IM star,HM
– 9 –
For the gas, we track the primordial and enriched gas masses separately. The primordial
gas is a combination of what was originally in the cloud core plus what gets accreted after
star formation begins, and it also includes the part of the stellar envelope debris that was not
converted into p-processed elements. We assume that massive stars make p-process elements
continuouslyandthatmixingfromrapidrotationputstheseelementsintothestellarenvelops
continuously. Thus we conceptually divide the envelope mass into an unprocessed fraction at
the original primordial abundance, and a completely processed, or enriched, fraction at the
abundance of fully processed material. The total envelope is a combination of these, giving
a partially-processed elemental abundance that comes from the dilution of fully processed
material by the mass that is still in an unprocessed form.
With these assumptions, the rate of change of the primordial (1st generation) gas mass
in the cloud core, M , is the increase from stellar debris and envelope accretion minus
gas,1
what goes into stars,
t f M˙ (t′) t−t′
M˙ (t) = debr star,HM (1−f (t′)) 1− dt′ (5)
gas,1 p
t t
Z0 evol ! (cid:20) evol (cid:21)
+R (t)−(1−f (t))M˙ (t).
acc p star
The first term in the integral is from ejection of stellar debris (equatorial disks, collisional
debris, Roche-lobe overflow). The term f is the average fraction of the mass of a high-
debr
mass star that is in the envelope and can be ejected. It equals 0.651 for M = 100 M
upper ⊙
and 0.568 for M = 300 M (Prantzos & Charbonnel 2006). Division by t indicates
upper ⊙ evol
that we assume this debris is ejected steadily over the evolution time of the massive star,
nominally assumed to be t = 3 Myr. The quantity f (t′) is the processed fraction in
evol p
the gas at time t′, and therefore also the processed fraction in stars that form at time t′,
assumingrapidmixing. Thus, 1−f (t′)istheunprocessedfractionofthemassofthestarthat
p
previously formed at t′. The last term, 1−([t−t′]/t ), tracks the remaining unprocessed
evol
fraction in the stellar envelope as the concentration of processed material increases linearly
with time. This linear increase assumes the p-process elements from the stellar core mix
into the stellar envelope at a steady rate. In addition to the integral that represents debris
output, the unprocessed gas mass also increases by accretion at the rate R . From these
acc
additions we subtract the unprocessed gas mass in the cloud core that goes into stars.
The rate of change of processed gas mass in the cloud core, M (2nd generation), is
gas,2
from the addition of stellar debris minus what goes into stars:
t f M˙ (t′) t−t′
M˙ (t) = debr star,HM (1−f (t′)) dt′ (6)
gas,2 p
t t
Z0 evol ! (cid:18) evol (cid:19)
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t f M˙ (t′)
+ debr star,HM f (t′)dt′ −f (t)M˙ (t).
p p star
t
Z0 evol !
The first integral represents the originally unprocessed mass fraction in the star when it
formed, (1−f (t′)), that came out as debris and became more and more contaminated with
p
time (as measured by [t−t′]/t ), and the second integral represents the return of originally
evol
processed mass (at fraction f ) into the cloud core. Note that the sum of the processed and
p
unprocessed gas mass rates from equations (5) and (6) equals t[f M (t′)/t ]dt′ +
0 debr HM evol
R −dM /dt,whichisthetotaldebrisrateplustheaccretionrateminusthestarformation
acc star R
rate.
Now we determine the masses of primordial and enriched gas in the star-forming cloud
core by integration,
t
M (t) = M˙ (t)dt (7)
gas,1 gas,1
Z0
t
M (t) = M˙ (t)dt, (8)
gas,2 gas,2
Z0
we combine these to get the total gas mass,
M = M +M , (9)
gas gas,1 gas,2
and we determine the mass fractions of enriched gas used above.
M
gas,2
f (t) = . (10)
p
M
gas
The low mass stars do not contribute to the above equations except as a long-term
sink for stellar mass. However, these stars are important for the observation of GCs today,
and this is where stellar ejection and evaporation come in. We assume here that only low
and intermediate mass stars leave the cluster by these processes, and we trace only the low
mass stellar loss because the intermediate mass stars will have disappeared by now anyway,
except as residual collapsed remnants. The high mass stars are assumed to segregate to
the center of the GC where essentially all of their p-processes elements are available for gas
contamination, as written in the above equations. Thus we need to model the escape of low
mass stars.
The discussion in Section 1 suggests that multi-star interactions and gas motions in the
GC core occasionally accelerate low mass stars up to escape speed or beyond. Thus the rate
of stellar ejection depends on the dynamical rate in the core, and this is directly proportional
to both the free-fall rate and the consumption rate in the basic model. This implies that